Question

which linear inequality is represented by the graph?
$\bigcirc\\ y\geq \frac{1}{3}x - 4$
$\bigcirc\\ y\leq \frac{1}{3}x - 4$
$\bigcirc\\ y\leq \frac{1}{3}x + 4$
$\bigcirc\\ y\geq \frac{1}{3}x + 4$

Explanation:

Step1: Find line equation (slope-intercept)

First, identify the y-intercept: the line crosses the y-axis at $(0, -4)$, so $b=-4$. Calculate the slope using two points, e.g., $(0, -4)$ and $(3, -3)$:
$$m=\frac{-3 - (-4)}{3 - 0}=\frac{1}{3}$$
The line equation is $y=\frac{1}{3}x - 4$.

Step2: Determine inequality direction

The line is solid, so the inequality includes equality. The shaded region is below the line, so we use $\leq$.

Answer:

$\boldsymbol{y \leq \frac{1}{3}x - 4}$ (Option B)